sci.op-research

dallasfairborn@yahoo.com wrote:
> I have a problem with mixed integer-valued and real-valued decision
> variables. The problem has a wide variety of constraints, but all are
> linear. The objective function is composed of a subset of the
> decision variables and can be expressed as the sum of products of
> these variables:
>
> Minimize Sum_i ( Product_j x_i_j)
>
> Here "Sum_i" denotes the summation over the index "i." All the
> decision variables in the objective function are real-valued and
> between 0 and 1. I tried to use a piecewise linear approximation
> (exponentiating the log of the product) for the objective function,
> but that won't work. I won't explain why, but I expect that this
> won't surprise anyone.
>
> I know this is a nonlinear programming problem, but I'm hoping that
> someone has experience with this kind of nonlinear program. I know
> that the objective function is a posynomial, but that's the extent of
> my nonlinear programming knowledge. I'd appreciate any advice, any
> suggested references, or just gut feelings for solving such a problem
> to optimality.
>
> If I can't find a way soon, I'll have to make use of one or more
> heuristic methods (GA, Simulated Annealing, Tabu, GRASP, etc.). I'd
> also appreciate any thoughts on heuristic methods for such problems.
> Thanks in advance for any help.

Could you elaborate on "wide variety of constraints".

If the linear constraints have a certain structure, you could be able to
rewrite the problem to a mixed integer geometric program, which can be,
if not efficiently, at least much more efficiently than a general MINLP.